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  <doc-data|<doc-title|Sampling Rate Reduction>>

  1. What is down sampling in the time domain?

  <\indent>
    With down sampling of 2, we would take every other sample skiping each
    one.
  </indent>

  2. What is the equation for down sampling ?

  <\equation*>
    x<rsub|d><around*|[|n|]>=x<around*|[|n M|]>
  </equation*>

  3. To avoid aliasing during down sampling, what should the signal be
  bandlimited to?

  <\indent>
    The bandwidth of the signal multiplied by M must be less than or equal to
    <math|\<pi\>>

    or <math|\<omega\> M \<leq\> \<pi\>>
  </indent>

  4. During down sampling of M, how does the bandwith of the signal change?

  <\indent>
    The bandwidth of the original signal is multiplied by M
  </indent>

  5. While down sampling of M, what happens to the height of the signal?\ 

  <\indent>
    The height is divided by M, during down sampling, M replicas of the
    original signal will be made at <math|<frac|2\<pi\>|M>>, so the signal
    strength is also divided by M. It is important to realize that when we
    sample a signal from continuous to discrete, the amplitude is already
    decreased by <math|<frac|1|T>>, if we then down sample it again, it would
    further decrease by M.
  </indent>

  6. What is the relationship between the digital signal
  <math|X<around*|(|e<rsup|j w>|)>> to its down sampled version
  <math|X<rsub|d><around*|(|e<rsup|j w>|)>> ?

  <\equation*>
    X<rsub|d><around*|(|e<rsup|j w>|)>= <frac|1|M><above|<below|<big|sum>|k=0>|M-1>X<around*|(|e<rsup|j<around*|(|w-2\<pi\>k|)>/M>|)>
  </equation*>

  7. To avoid aliasing, it would be ideal to pre-filter the signal with a low
  pass filter at what band?\ 

  <\equation*>
    \<omega\> = <frac|\<pi\>|M>
  </equation*>

  8. If a signal uses a low pass filter prior to the down sampling, what is
  this process called?\ 

  <\indent>
    Deciamation
  </indent>

  \;

  9. During the decimation process what should the magnitude of the filter
  be?

  <\indent>
    The magnitude should be 1. After sampling, the magnitude of the discrete
    signal becomes <math|<frac|1|T>> . From <math|<frac|1|T>>, the down
    sampling divides the magnitude by M, the new magnitude become
    <math|<frac|1|M T>>. Since this is already what the new magnitude should
    be, the low pass filter's magnitude, is equal to 1.\ 
  </indent>

  \;

  \;

  <doc-data|<doc-title|Sampling Rate Increase>>

  1. What is upsampling in the time domain?

  <\indent>
    If the upsampling is done by L, the distance between each sample is
    padded with L - 1 \ zeros. \ If it is an upsampling of 2, every other
    sample is now padded with 0.\ 
  </indent>

  2. What is the equation for down sampling ?

  <\equation*>
    x<rsub|d><around*|[|n|]>=x<around*|[|n /L|]>
  </equation*>

  3. Upsampling does not create extra copies, what does it have instead?

  <\indent>
    Upsampling creates images resulting from the higher copies moving down by
    L. Because of these images, a low pass filter is used to remove them
    after the upsampling.\ 
  </indent>

  4. During upsampling of L, how does the bandwith of the signal change?

  <\indent>
    The bandwidth of the original signal is divided by L
  </indent>

  5. While upsampling of L, what happens to the height of the signal?\ 

  <\indent>
    The height stays the same
  </indent>

  6. What is the relationship between the digital signal
  <math|X<around*|(|e<rsup|j w>|)>> to its upsampled version
  <math|X<rsub|d><around*|(|e<rsup|j w>|)>> ?

  <\equation*>
    X<rsub|u><around*|(|e<rsup|j w>|)>= X<around*|(|e<rsup|j w L>|)>
  </equation*>

  7. How do you remove the image from upsampling?

  <\indent>
    Use a low pass filter with <math|\<omega\>> with a cut off frequency of :
  </indent>

  <\equation*>
    \<omega\> = <frac|\<pi\>|L>
  </equation*>

  8. What is the cascade of an up-sampler with a low pass filter called?

  <\indent>
    Interpolator
  </indent>

  9. What happens in the time domain when we add a low pass filter to an up
  sampler?

  <\indent>
    The 0 padding between each sample becomes an interpolation between the
    points.
  </indent>

  10. What is the equation for a rational factor sampling rate change?

  <\equation*>
    x<rsub|r> = x<around*|[|n M/L|]>
  </equation*>

  11. What is the cascade image of a rational factor sampling rate change?

  <\indent>
    Starts with an up sampler following two low pass filters and then a down
    sampler.\ 
  </indent>

  12. The rational sampling rate change implies that the two filters can be
  combined in the middle. What is the bandwidth of the combined low pass
  filter?

  <\indent>
    The smaller of <math|<frac|\<pi\>|M>> or <math|<frac|\<pi\>|L>>
  </indent>

  \;

  13. Changing the period of T to <math|<frac|2|3>>T, what does this imply in
  terms of up and down sampler?

  <\indent>
    This implies up sampling of 3 and down sampling of 2.
  </indent>

  \;

  14. During the interpolation process what should the magnitude of the
  filter be?

  <\indent>
    The magnitude should be L. After sampling, the magnitude of the discrete
    signal becomes <math|<frac|1|T>> . From <math|<frac|1|T>>, the upsampling
    does not change the magnitude. The new magnitude stays at
    <math|<frac|1|T>>, when it should be <math|<frac|L|T>>. For this reason,
    the magnitude of the low pass filter for the up sampler should be L. Even
    when we have a rational sampling rate, we would use the magnitude of L.\ 
  </indent>

  \;

  \;

  <doc-data|<doc-title|Quantization Errors>>

  1. What is the quantization step size?\ 

  <\indent>
    The step size is the resolution of the quantizer, it is normally
    represented with the delta symbol <math|\<Delta\>>
  </indent>

  2. If the quantization resolution is <math|\<Delta\>> what is the maximum
  size of the quantization error ?

  <\indent>
    The maximum error goes from -<math|<frac|\<Delta\>|2>> to
    <math|<frac|\<Delta\>|2>>.\ 
  </indent>

  3. What is the equation for the expectation of a continuous uniform
  distribution?

  <\equation*>
    E<around*|(|x|)>=<big|int>x f<around*|(|x|)> \<partial\>x
  </equation*>

  4. What is the equation for the variance of a continuous uniform
  distribution?

  <\equation*>
    Var<around*|(|x|)>=E<around*|[|<around*|(|x<text|->u|)><rsup|2>|]>=<big|int>x<rsup|2>f<around*|(|x|)>\<partial\>x
    <text|--> <around*|[|<big|int>x f<around*|(|x|)>\<partial\>x|]><rsup|2>
  </equation*>

  5. What is the variance of the quantization error ?

  <\equation*>
    \<sigma\><rsup|2>=<frac|\<Delta\><rsup|2>|12>
  </equation*>

  6. How much does the signal to quantization error ratio improve for each
  extra bit we add?

  <\indent>
    6 dB
  </indent>

  7. What is a codeword?

  <\indent>
    The output of the encoder. A unique binary number corresponding to each
    quantization level.
  </indent>

  \;

  \;

  <doc-data|<doc-title|Multirate Signal Processing>>

  1. What are two multirate signal processing techniques?

  <\indent>
    First is the exchange before the up-down sampler with the filters.\ 

    Second is the polyphase decomposition technique
  </indent>

  2. When is exchange of up and down sampler useful?

  <\indent>
    They are used when the up or down sampling is very large. In those cases,
    since the bandwidth needs to be <math|<frac|\<pi\>|L>> or
    <math|<frac|\<pi\>|M> > , if L or M is very large, the low pass filter
    needs to have a very thin band. In these situations, we can cascade two
    low pass filters to replace the single one.
  </indent>

  3. Describe how the cascade system would work?

  <\indent>
    With two filters, the bandwidth of each can be
    <math|<frac|\<pi\>|M<rsub|1>>> and <math|<frac|\<pi\>|M<rsub|2>>>, but if
    we are trying to do it in a single stage, the band must be
    <math|<frac|\<pi\>|M<rsub|1>M<rsub|2>>>. This is obviously must thinner
    than either one. The cascade system can be visualized as 2 low pass
    filters following 2 down or up samplers.
  </indent>

  4. Describe how the polyphase implementation work?

  <\indent>
    It deinterleaves a signal. If it is 2 way, it splits every other. It then
    up samples and recombines them.\ 
  </indent>
</body>

<\initial>
  <\collection>
    <associate|language|american>
    <associate|page-type|letter>
  </collection>
</initial>